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# Full text of "Treatise On Analysis Vol-Ii"

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142       XIII   INTEGRATION

(13.9.18) Let fi be a bounded positive measure on X. Then for each & > 0 there
exists a compact subset KofX such that jj.(X - K) ^ e.

This is a particular case of (13.7.9).

(1 3.9.1 9) Suppose that \JL is bounded. Let (/,) be a uniformly bounded sequence
of measurable functions (i.e., sup ||/J < + oo) such that limfn(x) =/(*) exists

n                                                       H-+OO

for almost all x e X. Then f is integrable, and we have

(13.9.19.1)                      |/^=lim \fnd/i..

J             »•+» J

This is a particular case of (13.8.4).

Example

(13.9.20) Let JJL be a positive measure on X and let Y be a closed subspace
of X. It follows from the Tietze-Urysohn theorem (4.5.1 ) that every function
/e Jf K(Y) is the restriction to Y of a function g e <%£ (X) such that ||#|| = ||/||.
The function /Y = g<py , which is equal to/in Y and zero outside Y, is there-
fore ju-integrable (13.9.13), and |^(/Y)| g ||/|| • XSupp(/)). Write ,UY(/) =
K/Y) (which clearly depends only on/); then the discussion above shows that
/-» juy(/) is a positive measure on Y, which is said to be the measure induced
by IL on Y. From (12.7.8), the remarks above and (13.5.7), it follows im-
mediately that if /belongs to ,/(Y) and if as above we denote by/Y the func-
tion which is equal to/in Y and is zero in X — Y, then we have ju*(/) = /^*(/Y).
Hence we conclude (13.5.5) that this formula is valid for any mapping
/: Y -+ R (with/Y defined as before). Hence, finally, a function /: Y -+ E is
/iY-5ntegrable if and only if /Y is ^-integrable, and that in this case we have

(13.9.20.1)

f      - f

J  ^Y r

Likewise, a mapping u of Y into a topological space Z is /*Y-;measurable
if and only if the mapping u': X -> Z which extends u to X by giving u'(x) an
(arbitrary) constant value for x e X - Y, is /i-measurable.

Conversely, consider a closed subset Y of X and a positive measure v on
Y, and let \JL be the canonical extension of v to X (13.1.7). Clearly we have
v = uy. Also, if/e >(Y), then the function g which is equal to/on Y and
to +00 on X - Y belongs to J^(X), and we have v*(/) = ^(g\ It follows Ln is continuous.